2007/03/31 by Ileana Streinu, Louis Theran, Streinu, Ileana +1
Computer Science · Engineering · #05B35 #05C70 #05C85 #68R10 #Advanced Graph Theory Research #Advanced Materials and Mechanics #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.0704.0002
openalex publication_date 2007/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe a new algorithm, the (k,ℓ)-pebble game with colors, and use it obtain a characterization of the family of (k,ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k,ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and Westermann and Hendrickson.